Theorem mexval 35145

Description: The set of expressions, which are pairs whose first element is a typecode, and whose second element is a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mexval.k	⊢ 𝐾 = (mTC‘𝑇)
mexval.e	⊢ 𝐸 = (mEx‘𝑇)
mexval.r	⊢ 𝑅 = (mREx‘𝑇)

Assertion

Ref	Expression
mexval	⊢ 𝐸 = (𝐾 × 𝑅)

Proof of Theorem mexval

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mexval.e	. 2 ⊢ 𝐸 = (mEx‘𝑇)
2		fveq2 6902	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mTC‘𝑡) = (mTC‘𝑇))
3		mexval.k	. . . . . 6 ⊢ 𝐾 = (mTC‘𝑇)
4	2, 3	eqtr4di 2786	. . . . 5 ⊢ (𝑡 = 𝑇 → (mTC‘𝑡) = 𝐾)
5		fveq2 6902	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇))
6		mexval.r	. . . . . 6 ⊢ 𝑅 = (mREx‘𝑇)
7	5, 6	eqtr4di 2786	. . . . 5 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅)
8	4, 7	xpeq12d 5713	. . . 4 ⊢ (𝑡 = 𝑇 → ((mTC‘𝑡) × (mREx‘𝑡)) = (𝐾 × 𝑅))
9		df-mex 35130	. . . 4 ⊢ mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡)))
10		fvex 6915	. . . . 5 ⊢ (mTC‘𝑡) ∈ V
11		fvex 6915	. . . . 5 ⊢ (mREx‘𝑡) ∈ V
12	10, 11	xpex 7761	. . . 4 ⊢ ((mTC‘𝑡) × (mREx‘𝑡)) ∈ V
13	8, 9, 12	fvmpt3i 7015	. . 3 ⊢ (𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅))
14		xp0 6167	. . . . 5 ⊢ (𝐾 × ∅) = ∅
15	14	eqcomi 2737	. . . 4 ⊢ ∅ = (𝐾 × ∅)
16		fvprc 6894	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = ∅)
17		fvprc 6894	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mREx‘𝑇) = ∅)
18	6, 17	eqtrid 2780	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅)
19	18	xpeq2d 5712	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝐾 × 𝑅) = (𝐾 × ∅))
20	15, 16, 19	3eqtr4a 2794	. . 3 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅))
21	13, 20	pm2.61i 182	. 2 ⊢ (mEx‘𝑇) = (𝐾 × 𝑅)
22	1, 21	eqtri 2756	1 ⊢ 𝐸 = (𝐾 × 𝑅)

Syntax hints:  ¬ wn 3   = wceq 1533   ∈ wcel 2098  Vcvv 3473  ∅c0 4326   × cxp 5680  ‘cfv 6553  mTCcmtc 35107  mRExcmrex 35109  mExcmex 35110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-mex 35130

This theorem is referenced by:  mexval2  35146  msubff  35173  msubco  35174  msubff1  35199  mvhf  35201  msubvrs  35203